Question: You are given that $x$ is directly proportional to $y^3$, and $y$ is inversely proportional to $\sqrt{z}$.  If the value of $x$ is 3 when $z$ is $12$, what is the value of $x$ when $z$ is equal to $75$?  Express your answer as a common fraction.
Solution: By the definition of direct variation, we know that $x=my^3$ for some constant $m$. By the definition of inverse proportion, we know that $y=n/\sqrt{z}$ for some constant $n$.  Substituting for $y$ in the first expression, we can see that $x=\frac{mn^3}{(\sqrt{z})^3}=\frac{k}{z\sqrt{z}}$ or $xz\sqrt{z}=k$ for some constant $k$.  Substituting the given values, we can solve for $k$: $$xz\sqrt{z}=3\cdot 12\sqrt{12}=36\cdot 2\sqrt{3}=72\sqrt{3}=k$$Now, we can let $z=75$ and use the value of $k$ to solve for $x$: \begin{align*}
xz\sqrt{z}=x(75\sqrt{75})&=72\sqrt{3}\\
\Rightarrow\qquad x(75\cdot5\sqrt{3})&=72\sqrt{3}\\
\Rightarrow\qquad 375\sqrt{3}x&=72\sqrt{3}\\
\Rightarrow\qquad x&=72/375=\boxed{\frac{24}{125}}
\end{align*}